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ME521 Really Advanced Fluid Mechanics

Hwk 1c

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Computer Project--

DueFebruary 4,2003

A plastic toy rocket is propelled by a jet of water forced out thenozzle by compressed air. Your assignment is to create a model of arocket and determine the optimum initial water mass in the rocket. Thiswill require using the conservation of mass and momentum in an unsteadymanner.

Conservation of mass. The mass of the rocket changes continuously withtime as water leaves. Neglect the mass of the air.

Velocity at the nozzle exit

can be calculated from the Bernoulliequation. For simplicity, use the steady Bernoulli equation and ignoregravitational effects and the velocity of the air-water interfaceinside the rocket. The air pressure will change continuously as

thewater leaves and the air expands. We can assume that the expansion isisentropic (i.e., adiabatic and reversible).

Conservation of momentum. The acceleration of the rocket, as a functionof time, is determined from a momentum study in the vertical direction.For simplicity assume all the water inside the rocket has the samevelocity as the rocket itself. The momentum flux from the water is thedriving force and the acceleration of gravity opposes it. Neglect airdrag on the outside.

Numerical time marching. The equations are nonlinear with time-varyingcoefficients. The only way to solve them is to do it numerically. Wecan do this by a time-marching techniquecalled an explicit Eulertechnique.For example, consider the spring-mass-damper system

wherex

is the displacement of massm,

$b$ and $k$ are the damper andspring constants, and $f$ is a forcing term. We can solve this bymarching in time by forming a system of first order equations formedfrom the following Taylorseries equations:

To start, we sett=0

and sett

to a small value. All of the terms onthe right-hand side are known from initial conditions except for)which is determined from the

differential equation. Then the processis repeated with the terms on the right-hand side evaluated att=t

tofind the values fort=2t…

For the rocket problem, you will need(at least)three first-orderequations forz,z

andm. (A fourth equation for $Vnozzle

is needed ifyou decide to solve the unsteady Bernoulli equation. You will need touse small time steps until the water or pressure“runs out”. Then alarger time step is sufficient. The minimum information you will needis given below:

M_{rocket}= .0184 kg

Vrocket=74.9 x 10-6

m3

Pinit=5 atm

Dnozzle=.0055m

whereP

is considered gauge pressure. Your write-up should beapproximately 3 pages long (excluding figures and program listing oroutput) and contain the following items in an appropriate order: